Apparatus and method for recovery of three dimensional magnetic field from a magnetic detection system

ABSTRACT

A system for magnetic detection of an external magnetic field is disclosed. The system includes a nitrogen vacancy (NV) diamond material comprising a plurality of NV centers, a magnetic field generator that generates a magnetic field, a radio frequency (RF) excitation source that provides RF excitation, an optical excitation source that provides optical excitation, an optical detector that receives an optical signal emitted by the NV diamond material, and a controller. The controller is configured to calculate a control magnetic field, control the magnetic field generator to generate the control magnetic field, receive a light detection signal from the optical detector based on the optical signal due to the sum of the generated control magnetic field and the external magnetic field, store measurement data based on the received light detection signal, and calculate a vector of the external magnetic field based on the stored measurement data.

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

The present application claims the benefit of priority from U.S. Provisional Patent Application No. 62/112,071, filed Feb. 4, 2015, which is incorporated herein by reference in its entirety. This application is related to co-pending U.S. patent application filed Jan. 21, 2016, titled “APPARATUS AND METHOD FOR ESTIMATING ABSOLUTE AXES' ORIENTATIONS FOR A MAGNETIC DETECTION SYSTEM”, Ser. No. 15/003,704, which is incorporated by reference herein in its entirety.

TECHNICAL FIELD

The present disclosure generally relates to magnetometers, and more particularly, to apparatuses and methods for the recovery of a three-dimensional (3-D) magnetic field using a single fixed diamond nitrogen vacancy sensor.

BACKGROUND

A number of industrial applications including, but not limited to, medical devices, communication devices, and navigation systems, as well as scientific areas such as physics and chemistry can benefit from magnetic detection and imaging. Many advanced magnetic imaging systems can operate in limited conditions, for example, high vacuum and/or cryogenic temperatures, which can make them inapplicable for imaging applications that require ambient conditions. Furthermore, small size, weight and power (SWAP) magnetic sensors of moderate sensitivity, vector accuracy, and bandwidth are valuable in many applications.

Atomic-sized nitrogen vacancy (NV) centers in diamond lattices have been shown to have excellent sensitivity for magnetic field measurement and enable fabrication of small magnetic sensors that can readily replace existing-technology (e.g., Hall-effect, SERF, SQUID) systems and devices. The sensing capabilities of diamond NV center sensors are maintained in room temperature and atmospheric pressure and these sensors can be even used in liquid environments (e.g., for biological imaging). Measurement of 3-D vector magnetic fields via diamond NV sensing may be beneficial across a very broad range of applications including communications, geological sensing, navigation, and attitude determination.

Currently, methods to measure the full 3-D vector of an external magnetic field are cumbersome and time-consuming. For example, such methods require the use of multiple sensors, each dedicated to measuring one direction of the 3-D vector, which are combined to determine the full magnetic field vector. Other methods utilize a single sensor dedicated to measuring one direction of the magnetic field vector at a time, thereby increasing the time required to determine the full magnetic field vector.

SUMMARY

According to certain embodiments, a system for magnetic detection of an external magnetic field may include a nitrogen vacancy (NV) diamond material comprising a plurality of NV centers, a magnetic field generator configured to generate a magnetic field, a radio frequency (RF) excitation source configured to provide RF excitation to the NV diamond material, an optical excitation source configured to provide optical excitation to the NV diamond material, an optical detector configured to receive an optical signal emitted by the NV diamond material, the optical signal being a fluorescence intensity having a plurality of reduced responses across a frequency range of the RF excitation, and a controller. The controller may be configured to calculate a control magnetic field that separates the plurality of reduced responses in the optical signal emitted by the NV diamond material, control the magnetic field generator to generate the control magnetic field, receive a light detection signal from the optical detector based on the optical signal emitted by the NV diamond material due to the sum of the generated control magnetic field and the external magnetic field, store measurement data based on the received light detection signal, and calculate a vector of the external magnetic field based on the stored measurement data.

According to one aspect, a controller may calculate a control magnetic field that equally separates the plurality of reduced responses.

According to one aspect, a controller may calculate a control magnetic field that maximally separates the plurality of reduced responses.

According to one aspect, a controller may be further configured to calculate an orientation of the NV diamond material.

According to one aspect, a controller may be configured to calculate the orientation of the NV diamond material based on a rotation and/or reflection of a standard orientation of the NV diamond material.

According to one aspect, a controller may be configured to assign a plurality of sign values to the measurement data such that a sum of the measurement data approaches zero.

According to one aspect, a controller may be configured to assign a positive sign value to the largest and smallest measurement data, and a negative sign value to the second-largest and third-largest measurement data.

According to one aspect, a controller may be configured to assign a positive sign value to the largest measurement data, and a negative sign value to the second-largest, third-largest, and smallest measurement data.

According to other embodiments, a system for magnetic detection may include a nitrogen vacancy (NV) diamond material comprising a plurality of NV centers, a magnetic field generator configured to generate a magnetic field, a radio frequency (RF) excitation source configured to provide RF excitation to the NV diamond material, an optical excitation source configured to provide optical excitation to the NV diamond material, an optical detector configured to receive an optical signal emitted by the NV diamond material, the optical signal being a fluorescence intensity having a plurality of reduced responses across a frequency range of the RF excitation, and a controller. The controller may be configured to calculate a control magnetic field that separates the plurality of reduced responses in the optical signal emitted by the NV diamond material, and control the magnetic field generator to generate the control magnetic field.

According to one aspect, a controller may calculate a control magnetic field that equally separates the plurality of reduced responses.

According to one aspect, a controller may calculate a control magnetic field that maximally separates the plurality of reduced responses.

According to one aspect, a controller may be further configured to receive a light detection signal from the optical detector based on the optical signal emitted by the NV diamond material due to the sum of the generated control magnetic field and the external magnetic field, store measurement data based on the received light detection signal, and calculate a vector of the external magnetic field based on the stored measurement data.

According to one aspect, a system may further include a pivot assembly. The magnetic field generator may be affixed to the pivot assembly, the pivot assembly may be configured to position the magnetic field generator to a predetermined orientation such that the magnetic field generator generates the control magnetic field, and the controller may be further configured to control the pivot assembly.

According to other embodiments, a system for magnetic detection of an external magnetic field may include a nitrogen vacancy (NV) diamond material comprising a plurality of NV centers, a radio frequency (RF) excitation source configured to provide RF excitation to the NV diamond material, an optical excitation source configured to provide optical excitation to the NV diamond material, an optical detector configured to receive an optical signal emitted by the NV diamond material, the optical signal being a fluorescence intensity having a plurality of reduced responses across a frequency range of the RF excitation, a magnetic field generator configured to generate a control magnetic field, the control magnetic field being configured to separate the plurality of reduced responses in the optical signal emitted by the NV diamond material, and a controller. The controller may be configured to receive a light detection signal from the optical detector based on the optical signal emitted by the NV diamond material due to the sum of the control magnetic field and the external magnetic field, store measurement data based on the received light detection signal, and calculate a vector of the external magnetic field based on the stored measurement data.

According to one aspect, a control magnetic field may be configured to equally separate the plurality of reduced responses.

According to one aspect, a control magnetic field may be configured to maximally separate the plurality of reduced responses.

According to one aspect, a controller may be further configured to calculate an orientation of the NV diamond material.

According to one aspect, a controller may be configured to calculate the orientation of the NV diamond material based on a rotation and/or reflection of a standard orientation of the NV diamond material.

According to one aspect, a magnetic field generator may be a permanent magnet.

According to one aspect, a system may further include a pivot assembly. The magnetic field generator may be affixed to the pivot assembly, the pivot assembly may be configured to position the magnetic field generator to a predetermined orientation such that the magnetic field generator generates the control magnetic field, and the controller may be further configured to control the pivot assembly.

According to other embodiments, a system for magnetic detection of an external magnetic field may include a nitrogen vacancy (NV) diamond material comprising a plurality of NV centers, a radio frequency (RF) excitation source configured to provide RF excitation to the NV diamond material, an optical excitation source configured to provide optical excitation to the NV diamond material, and an optical detector configured to receive an optical signal emitted by the NV diamond material, the optical signal being a fluorescence intensity having a plurality of reduced responses across a frequency range of the RF excitation. The system may further include a magnetic field generator affixed to a pivot assembly, the pivot assembly being configured to position the magnetic field generator to a predetermined orientation such that the magnetic field generator generates a control magnetic field that separates the plurality of reduced responses in the optical signal emitted by the NV diamond material, and a controller. The controller may be configured to control the pivot assembly to position the magnetic field generator to the predetermined orientation to generate the control magnetic field, receive a light detection signal from the optical detector based on the optical signal emitted by the NV diamond material due to the sum of the control magnetic field and the external magnetic field, store measurement data based on the received light detection signal, and calculate a vector of the external magnetic field based on the stored measurement data.

According to other embodiments, a method for detecting an external magnetic field applied on a nitrogen vacancy (NV) diamond material comprising a plurality of NV centers may include providing radio frequency (RF) excitation to the NV diamond material, providing optical excitation to the NV diamond material, detecting an optical signal emitted by the NV diamond material, the optical signal being a fluorescence intensity having a plurality of reduced responses across a frequency range of the RF excitation, calculating a control magnetic field that separates the plurality of reduced responses in the optical signal emitted by the NV diamond material, applying the control magnetic field to the NV diamond material, receiving a light detection signal based on the optical signal emitted by the NV diamond material due to the sum of the generated control magnetic field and the external magnetic field, storing measurement data based on the received light detection signal, and calculating a vector of the external magnetic field based on the stored measurement data.

According to one aspect, a control magnetic field may separate the plurality of reduced responses.

According to one aspect, a control magnetic field may maximally separate the plurality of reduced responses.

According to one aspect, a method may further include calculating an orientation of the NV diamond material.

According to one aspect, a method may further include calculating the orientation of the NV diamond material is based on a rotation and/or reflection of a standard orientation of the NV diamond material.

According to other embodiments, a system for magnetic detection of an external magnetic field may include a magneto-optical defect center material comprising a plurality of magneto-optical defect centers, a magnetic field generator configured to generate a magnetic field, a radio frequency (RF) excitation source configured to provide RF excitation to the magneto-optical defect center material, an optical excitation source configured to provide optical excitation to the magneto-optical defect center material, an optical detector configured to receive an optical signal emitted by the magneto-optical defect center material, the optical signal being a fluorescence intensity having a plurality of reduced responses across a frequency range of the RF excitation, and a controller. The controller may be configured to calculate a control magnetic field that separates the plurality of reduced responses in the optical signal emitted by the magneto-optical defect center material, control the magnetic field generator to generate the control magnetic field, receive a light detection signal from the optical detector based on the optical signal emitted by the magneto-optical defect center material due to the sum of the generated control magnetic field and the external magnetic field, store measurement data based on the received light detection signal, and calculate a vector of the external magnetic field based on the stored measurement data.

According to one aspect, a controller may calculate a control magnetic field that equally separates the plurality of reduced responses.

According to one aspect, a controller may calculate a control magnetic field that maximally separates the plurality of reduced responses.

According to one aspect, a controller may be further configured to calculate an orientation of the magneto-optical defect center material.

According to one aspect, a controller may be configured to calculate the orientation of the magneto-optical defect center material based on a rotation and/or reflection of a standard orientation of the magneto-optical defect center material.

According to one aspect, a magneto-optical defect center material may be a nitrogen vacancy (NV) diamond material comprising a plurality of NV centers.

According to other embodiments, a system for detecting an external magnetic field acting on a nitrogen vacancy (NV) diamond material comprising a plurality of NV centers may include means for providing radio frequency (RF) excitation to the NV diamond material, means for providing optical excitation to the NV diamond material, means for detecting an optical signal emitted by the NV diamond material, the optical signal being a fluorescence intensity having a plurality of reduced responses across a frequency range of the RF excitation, means for generating a control magnetic field that separates the plurality of reduced responses in the optical signal emitted by the NV diamond material, means for receiving a light detection signal based on the optical signal emitted by the NV diamond material due to the sum of the generated control magnetic field and the external magnetic field, means for storing measurement data based on the received light detection signal, and means for calculating a vector of the external magnetic field based on the stored measurement data.

According to one aspect, a system may further include means for calculating an orientation of the NV diamond material.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates one orientation of an NV center in a diamond lattice.

FIG. 2 is an energy level diagram showing energy levels of spin states for the NV center.

FIG. 3 is a schematic diagram illustrating a conventional NV center magnetic sensor system.

FIG. 4 is a graph illustrating the fluorescence as a function of an applied RF frequency of an NV center along a given direction for a zero magnetic field.

FIG. 5 is a graph illustrating the fluorescence as a function of an applied RF frequency for four different NV center orientations for a non-zero magnetic field.

FIG. 6 is a schematic diagram illustrating a magnetic field detection system according to an embodiment.

FIG. 7A is a unit cell diagram of the crystal structure of a diamond lattice having a standard orientation.

FIG. 7B is a unit cell diagram of the crystal structure of a diamond lattice having an unknown orientation.

FIG. 8 is a schematic diagram illustrating a step in a method for determining the unknown orientation of the diamond lattice of FIG. 7B.

FIG. 9 is a flowchart illustrating a sign recovery method for the method for determining the unknown orientation of the diamond lattice of FIG. 7B.

FIG. 10 is a schematic diagram illustrating a step in the method for determining the unknown orientation of the diamond lattice of FIG. 7B.

FIG. 11 is a flowchart illustrating a method for recovering a three-dimensional magnetic field on the NV center magnetic sensor system.

DETAILED DESCRIPTION

The present disclosure relates to apparatuses and methods for recovering a full three-dimensional magnetic field from a single diamond NV center sensor system. The method utilizes high accuracy estimation methods for determining the axes orientation of the diamond lattice relative to the sensor system, establishing an optimal bias magnetic field to separate the four Lorentzian responses of the NV center to accurately measure RF shifts due to the external magnetic field, and computing an accurate estimate of the external magnetic field on the sensor system. The apparatuses and methods may utilize only a single sensor from which the full magnetic field vector may be recovered from simple measurements of the Lorentzian responses of the NV center.

The NV Center, its Electronic Structure, and Optical and RF Interaction

The NV center in a diamond comprises a substitutional nitrogen atom in a lattice site adjacent a carbon vacancy as shown in FIG. 1. The NV center may have four orientations, each corresponding to a different crystallographic orientation of the diamond lattice.

The NV center may exist in a neutral charge state or a negative charge state. Conventionally, the neutral charge state uses the nomenclature NV⁰, while the negative charge state uses the nomenclature NV, which is adopted in this description.

The NV center has a number of electrons, including three unpaired electrons, each one from the vacancy to a respective of the three carbon atoms adjacent to the vacancy, and a pair of electrons between the nitrogen and the vacancy. The NV center, which is in the negatively charged state, also includes an extra electron.

The NV center has rotational symmetry, and as shown in FIG. 2, has a ground state, which is a spin triplet with ³A₂ symmetry with one spin state m_(s)=0, and two further spin states m_(s)=+1, and m_(s)=−1. In the absence of an external magnetic field, the m_(s)=±1 energy levels are offset from the m_(s)=0 due to spin-spin interactions, and the m_(s)=±1 energy levels are degenerate, i.e., they have the same energy. The m_(s)=0 spin state energy level is split from the m_(s)=±1 energy levels by an energy of 2.87 GHz for a zero external magnetic field.

Introducing an external magnetic field with a component along the NV axis lifts the degeneracy of the m_(s)=±1 energy levels, splitting the energy levels m_(s)=±1 by an amount 2gμ_(B)Bz, where g is the g-factor, μ_(B) is the Bohr magneton, and Bz is the component of the external magnetic field along the NV axis. This relationship is correct to a first order and inclusion of higher order corrections is a straightforward matter and should not affect the computational and logic steps in the systems and methods described below.

The NV center electronic structure further includes an excited triplet state ³E with corresponding m_(s)=0 and m_(s)=±1 spin states. The optical transitions between the ground state ³A₂ and the excited triplet ³E are predominantly spin conserving, meaning that the optical transitions are between initial and final states that have the same spin. For a direct transition between the excited triplet ³E and the ground state ³A₂, a photon of red light is emitted with a photon energy corresponding to the energy difference between the energy levels of the transitions.

There is, however, an alternative non-radiative decay route from the triplet ³E to the ground state ³A_(z) via intermediate electron states, which are thought to be intermediate singlet states A, E with intermediate energy levels. Significantly, the transition rate from the m_(s)=±1 spin states of the excited triplet ³E to the intermediate energy levels is significantly greater than the transition rate from the m_(s)=0 spin state of the excited triplet ³E to the intermediate energy levels. The transition from the singlet states A, E to the ground state triplet ³A₂ predominantly decays to the m_(s)=0 spin state over the m_(s)=±1 spins states. These features of the decay from the excited triplet ³E state via the intermediate singlet states A, E to the ground state triplet ³A₂ allow that if optical excitation is provided to the system, the optical excitation will eventually pump the NV center into the m_(s)=0 spin state of the ground state ³A₂. In this way, the population of the m_(s)=0 spin state of the ground state ³A₂ may be “reset” to a maximum polarization determined by the decay rates from the triplet ³E to the intermediate singlet states.

Another feature of the decay is that the fluorescence intensity due to optically stimulating the excited triplet ³E state is less for the m_(s)=±1 states than for the m_(s)=0 spin state. This is so because the decay via the intermediate states does not result in a photon emitted in the fluorescence band, and because of the greater probability that the m_(s)=±1 states of the excited triplet ³E state will decay via the non-radiative decay path. The lower fluorescence intensity for the m_(s)=±1 states than for the m_(s)=0 spin state allows the fluorescence intensity to be used to determine the spin state. As the population of the m_(s)=±1 states increases relative to the m_(s)=0 spin, the overall fluorescence intensity will be reduced.

The NV Center, or Magneto-Optical Defect Center, Magnetic Sensor System

FIG. 3 is a schematic diagram illustrating a conventional NV center magnetic sensor system 300 that uses fluorescence intensity to distinguish the m_(s)=±1 states, and to measure the magnetic field based on the energy difference between the m_(s)=+1 state and the m_(s)=−1 state. The system 300 includes an optical excitation source 310, which directs optical excitation to an NV diamond material 320 with NV centers. The system further includes an RF excitation source 330, which provides RF radiation to the NV diamond material 320. Light from the NV diamond may be directed through an optical filter 350 to an optical detector 340.

The RF excitation source 330 may be a microwave coil, for example. The RF excitation source 330, when emitting RF radiation with a photon energy resonant with the transition energy between ground m_(s)=0 spin state and the m_(s)=+1 spin state, excites a transition between those spin states. For such a resonance, the spin state cycles between ground m_(s)=0 spin state and the m_(s)=+1 spin state, reducing the population in the m_(s)=0 spin state and reducing the overall fluorescence at resonances. Similarly, resonance occurs between the m_(s)=0 spin state and the m_(s)=−1 spin state of the ground state when the photon energy of the RF radiation emitted by the RF excitation source is the difference in energies of the m_(s)=0 spin state and the m_(s)=−1 spin state, or between the m_(s)=0 spin state and the m_(s)=+1 spin state, there is a decrease in the fluorescence intensity.

The optical excitation source 310 may be a laser or a light emitting diode, for example, which emits light in the green, for example. The optical excitation source 310 induces fluorescence in the red, which corresponds to an electronic transition from the excited state to the ground state. Light from the NV diamond material 320 is directed through the optical filter 350 to filter out light in the excitation band (in the green, for example), and to pass light in the red fluorescence band, which in turn is detected by the detector 340. The optical excitation light source 310, in addition to exciting fluorescence in the diamond material 320, also serves to reset the population of the m_(s)=0 spin state of the ground state ³A₂ to a maximum polarization, or other desired polarization.

For continuous wave excitation, the optical excitation source 310 continuously pumps the NV centers, and the RF excitation source 330 sweeps across a frequency range that includes the zero splitting (when the m_(s)=±1 spin states have the same energy) energy of 2.87 GHz. The fluorescence for an RF sweep corresponding to a diamond material 320 with NV centers aligned along a single direction is shown in FIG. 4 for different magnetic field components Bz along the NV axis, where the energy splitting between the m_(s)=−1 spin state and the m_(s)=+1 spin state increases with Bz. Thus, the component Bz may be determined. Optical excitation schemes other than continuous wave excitation are contemplated, such as excitation schemes involving pulsed optical excitation, and pulsed RF excitation. Examples of pulsed excitation schemes include Ramsey pulse sequence, and spin echo pulse sequence.

In general, the diamond material 320 will have NV centers aligned along directions of four different orientation classes. FIG. 5 illustrates fluorescence as a function of RF frequency for the case where the diamond material 320 has NV centers aligned along directions of four different orientation classes. In this case, the component Bz along each of the different orientations may be determined. These results, along with the known orientation of crystallographic planes of a diamond lattice, allow not only the magnitude of the external magnetic field to be determined, but also the direction of the magnetic field.

While FIG. 3 illustrates an NV center magnetic sensor system 300 with NV diamond material 320 with a plurality of NV centers, in general, the magnetic sensor system may instead employ a different magneto-optical defect center material, with a plurality of magneto-optical defect centers. The electronic spin state energies of the magneto-optical defect centers shift with magnetic field, and the optical response, such as fluorescence, for the different spin states is not the same for all of the different spin states. In this way, the magnetic field may be determined based on optical excitation, and possibly RF excitation, in a corresponding way to that described above with NV diamond material.

FIG. 6 is a schematic diagram of a system 600 for a magnetic field detection system according to an embodiment. The system 600 includes an optical excitation source 610, which directs optical excitation to an NV diamond material 620 with NV centers, or another magneto-optical defect center material with magneto-optical defect centers. An RF excitation source 630 provides RF radiation to the NV diamond material 620.

As shown in FIG. 6, a first magnetic field generator 670 generates a magnetic field, which is detected at the NV diamond material 620. The first magnetic field generator 670 may be a permanent magnet positioned relative to the NV diamond material 620, which generates a known, uniform magnetic field (e.g., a bias or control magnetic field) to produce a desired fluorescence intensity response from the NV diamond material 620. In some embodiments, a second magnetic field generator 675 may be provided and positioned relative to the NV diamond material 620 to provide an additional bias or control magnetic field. The second magnetic field generator 675 may be configured to generate magnetic fields with orthogonal polarizations. In this regard, the second magnetic field generator 675 may include one or more coils, such as a Helmholtz coil. The coils may be configured to provide relatively uniform magnetic fields at the NV diamond material 620 and each may generate a magnetic field having a direction that is orthogonal to the direction of the magnetic field generated by the other coils. In some embodiments, only the first magnetic field generator 670 may be provided to generate the bias magnetic field. Alternatively, only the second magnetic field generator 675 may be provided to generate the bias magnetic field.

The system 600 further includes a controller 680 arranged to receive a light detection signal from the optical detector 640 and to control the optical excitation source 610, the RF excitation source 630, and the second magnetic field generator 675. The controller may be a single controller, or multiple controllers. For a controller including multiple controllers, each of the controllers may perform different functions, such as controlling different components of the system 600. The second magnetic field generator 675 may be controlled by the controller 680 via an amplifier 660, for example.

The RF excitation source 630 may be a microwave coil, for example. The RF excitation source 630 is controlled to emit RF radiation with a photon energy resonant with the transition energy between the ground m_(s)=0 spin state and the m_(s)=±1 spin states as discussed above with respect to FIG. 3.

The optical excitation source 610 may be a laser or a light emitting diode, for example, which emits light in the green, for example. The optical excitation source 610 induces fluorescence in the red from the NV diamond material 620, where the fluorescence corresponds to an electronic transition from the excited state to the ground state. Light from the NV diamond material 620 is directed through the optical filter 650 to filter out light in the excitation band (in the green, for example), and to pass light in the red fluorescence band, which in turn is detected by the optical detector 640. The optical excitation light source 610, in addition to exciting fluorescence in the NV diamond material 620, also serves to reset the population of the m_(s)=0 spin state of the ground state ³A₂ to a maximum polarization, or other desired polarization.

The controller 680 is arranged to receive a light detection signal from the optical detector 640 and to control the optical excitation source 610, the RF excitation source 630, and the second magnetic field generator 675. The controller may include a processor 682 and a memory 684, in order to control the operation of the optical excitation source 610, the RF excitation source 630, and the second magnetic field generator 675. The memory 684, which may include a nontransitory computer readable medium, may store instructions to allow the operation of the optical excitation source 610, the RF excitation source 630, and the second magnetic field generator 675 to be controlled. That is, the controller 680 may be programmed to provide control.

Axes of the Diamond Crystal Lattice

In deriving the total magnetic field vector impinging on the system 600 from the measurements obtained by the intensity response produced by the NV diamond material 620, it is desirable to establish the orientation of the axes of the diamond lattice of the NV diamond material 620 to allow for the accurate recovery of the magnetic field vector and maximize signal-to-noise information. However, as discussed above, the NV diamond material 620 may be arbitrarily oriented and, thus, have axes in an unknown orientation. Thus, in such a case, the controller 680 may be configured to compute an accurate estimation of the true orientation of the NV diamond lattice, which can be performed on-site as a calibration method prior to use. This information can be subsequently used to accurately recover the full vector information of an unknown external magnetic field acting on the system 600.

To begin, a desired geospatial coordinate reference frame relative to the system 600 by which measurement of the total magnetic field vector will take place is established. As shown in FIGS. 7A and 7B, a Cartesian reference frame having {x, y, z} orthogonal axes may be used, but any arbitrary reference frame and orientation may be used. FIG. 7A shows a unit cell 100 of a diamond lattice having a “standard” orientation. The axes of the diamond lattice will fall along four possible directions. Thus, the four axes in a standard orientation relative to the desired coordinate reference frame may be defined as unit vectors corresponding to:

$\begin{matrix} {{a_{S,1} = {\frac{1}{\sqrt{3}}\begin{bmatrix} {- 1} & {- 1} & 1 \end{bmatrix}}^{T}}{a_{S,2} = {\frac{1}{\sqrt{3}}\begin{bmatrix} {- 1} & 1 & {- 1} \end{bmatrix}}^{T}}{a_{S,3} = {\frac{1}{\sqrt{3}}\begin{bmatrix} 1 & {- 1} & {- 1} \end{bmatrix}}^{T}}{a_{S,4} = {\frac{1}{\sqrt{3}}\begin{bmatrix} 1 & 1 & 1 \end{bmatrix}}^{T}}} & (1) \end{matrix}$

For simplicity, the four vectors of equation (1) may be represented by a single matrix A_(S), which represents the standard orientation of the unit cell 100: A _(S) =[a _(S,1) a _(S,2) a _(S,3) a _(S,4)]  (2)

The angle between axis i and axis j may also be given by the (i, j)^(th) row of the following:

$\begin{matrix} {{\cos^{- 1}\left( {A_{S}^{T}A_{S}} \right)} = {\quad{{\cos^{- 1}\left\lbrack \begin{matrix} 1 & {- \frac{1}{3}} & {- \frac{1}{3}} & {- \frac{1}{3}} \\ {- \frac{1}{3}} & 1 & {- \frac{1}{3}} & {- \frac{1}{3}} \\ {- \frac{1}{3}} & {- \frac{1}{3}} & 1 & {- \frac{1}{3}} \\ {- \frac{1}{3}} & {- \frac{1}{3}} & {- \frac{1}{3}} & 1 \end{matrix} \right\rbrack} \approx {\quad\left\lbrack \begin{matrix} {0{^\circ}} & {109.47{^\circ}} & {109.47{^\circ}} & {109.47{^\circ}} \\ {109.47{^\circ}} & {0{^\circ}} & {109.47{^\circ}} & {109.47{^\circ}} \\ {109.47{^\circ}} & {109.47{^\circ}} & {0{^\circ}} & {109.47{^\circ}} \\ {109.47{^\circ}} & {109.47{^\circ}} & {109.47{^\circ}} & {109.47{^\circ}} \end{matrix} \right\rbrack}}}} & (3) \end{matrix}$

FIG. 7B is a unit cell 100′ that represents an arbitrarily placed NV diamond material having unknown axes orientation with respect to the coordinate reference frame. By defining the standard orientation matrix A_(S) with reference to the established coordinate reference frame, the arbitrary orientation shown in FIG. 7B may be obtained through a rotation and/or reflection of the standard orientation matrix. This can be achieved by applying a transformation matrix R, which is defined as a general 3×3 matrix representing the three-dimensional, orthogonal Cartesian space and is, at this stage, unknown. The transformation matrix may be used to obtain our desired matrix A as follows: A=RA _(S)  (4) Deriving the Total Magnetic Field Vector

As described above with reference to FIGS. 3-5, the total magnetic field acting on the system 600 may be measured fluorescently. These measurements may be modeled as a linear system from which the total magnetic field impinging on the sensor may be determined: m=|A ^(T) b+n|  (5)

Here, bε

^(3×1) represents the magnetic field vector acting inside the sensor system, expressed in Cartesian coordinates relative to the coordinate reference frame; A^(T)b represents the projection of the magnetic field vector onto each of the four, arbitrarily-placed NV center diamond lattice axes; nε

^(4×1) represents the sensor noise vector; and mε

^(4×1) represents the measurement vector, where the i^(th) element represents the estimated projection of the magnetic field onto the sensor axis i. In terms of units, it is assumed that the measurement vector has been converted from the DNV sensor's native units (in terms of microwave resonance frequency) into the units of magnetic field strength. Furthermore, the term |A^(T)b+n| represents the element-wise absolute value of A^(T)b+n, rather than its determinant.

Given the linear model for the magnetic field measurement of equation (5) a least squares estimate of the total magnetic field acting on the system 600 may be given by: {circumflex over (b)}=(A ^(T))⁺ m  (6)

In the above equation, the +superscript denotes the Moore-Penrose pseudoinverse. Because the three four-element columns of A^(T) are linearly independent, equation (6) may be rewritten as: {circumflex over (b)}=(AA ^(T))⁻¹ Am =(RA _(S) A _(S) ^(T) R ^(T))⁻¹ Am =(4/3RIR ^(T))⁻¹ Am =0.75(RR ^(T))⁻¹ Am  (7)

In equation (7), A_(S)A_(S) ^(T)=4/3I (established in more detail below) has been substituted. Because R is an orthogonal matrix, equation (7) can be reduced to: {circumflex over (b)}=0.75(I)⁻¹ Am=0.75Am  (8)

In equations (7)-(8), it was assumed that all the measurements were weighted equally. If, however, some of the axes have less variance in their measurements or are preferred for other reasons, then different weightings may be used for each of the axes for a more optimal least squares estimate. If wε

^(4×1) represents the positive weights for each of the measurements and W=diag(w), then the weighted least-squares formulation for the total magnetic field may be written as: {circumflex over (b)}=argmin_(bε)

_(3×1) ∥W ^(1/2)(A ^(T) {circumflex over (b)}−m)∥₂  (9)

Based on equation (9), the generalized least squares solution of equation (6) may now be written as: {circumflex over (b)}=(W ^(1/2) A ^(T))⁺ W ^(1/2) m=(AWA ^(T))⁻¹ AWm  (10)

For a perfect NV diamond material 620 having no defects (e.g., lattice misalignments, impurities, etc.) and no sensor noise, {circumflex over (b)} should be equal to b. However, in an imperfect system, it is possible to utilize a performance metric to determine the error associated with the measurement. One possible metric that may be used is a 2-norm of the residual vector minimized by the least squares solution. This metric γ may be given by: γ=∥A ^(T) {circumflex over (b)}−m∥ ₂ =∥A ^(T)(AA ^(T))⁻¹ Am−m∥ ₂ =∥(A ^(T)(AA ^(T))⁻¹ A−I)m∥ ₂  (11)

Because the residual vector is proportional to the measurement amplitude, the magnitude of the true magnetic field may be used to normalize the metric to give a consistent metric even in the presence of a changing true magnetic field:

$\begin{matrix} {\gamma^{\prime} = \frac{{{\left( {{{A^{T}\left( {AA}^{T} \right)}^{- 1}A} - 1} \right)m}}_{2}}{{b}_{2}}} & (12) \end{matrix}$

If the true magnetic field is not known, the measurement vector magnitude may be used to normalize the metric:

$\begin{matrix} {\gamma^{''} = \frac{{{\left( {{{A^{T}\left( {AA}^{T} \right)}^{- 1}A} - 1} \right)m}}_{2}}{{m}_{2}}} & (13) \end{matrix}$ Estimation of Absolute Axes' Orientation in the NV Diamond Material

By simple substitution of equation (4) into equation (5), the measurement obtained by the system 600 may be represented in terms of the standard orientation matrix: m=|A ^(T) b+n|=|(RA _(S))^(T) b+n|  (14)

As described above, a permanent magnet (e.g., the first magnetic field generator 670) and/or coils (e.g., the second magnetic field generator 675) may be used to adequately separate out the Lorentzian dips that correspond to the magnetic field measurements along each diamond axis. However, at this point, the orientations of the sensor's axes are unknown. Thus, the required bias or control magnetic field, defined as b_(bias), that will produce the desired dip separation is unknown.

As will be described in more detail below, there exists a plurality of b_(bias) vectors that can equally separate out the four Lorentzian dips for adequate measurement purposes. Moreover, for the purposes of determining the unknown orientation of the diamond lattice, it is not necessary to precisely place or apply the bias magnetic field that will result in perfectly equal dip separation, which may be more appropriate during field measurement of an external magnetic field. In this case, any b_(bias) vector that sufficiently separates the four dips may suffice for the determination of the unknown orientation of the diamond lattice, thus increasing the viable b_(bias) vectors appropriate for this step. Sufficient spectral dip separation, however, may depend on the width of the dips and the planned magnitude of the calibration magnetic fields (described below). The width of the dips varies, depending on diamond composition and sensor laser and/or RF excitation mechanisms. Based on the resulting widths due to inherent sensor characteristics, the magnitude and orientation should be sufficient to ensure that the anticipated maximum spectral shifts that will occur due to the calibration tests will maintain sufficient separation between neighboring Lorentzian dips.

FIG. 8 shows a step for determining a viable b_(bias) vector field. As shown in FIG. 8, the first magnetic field generator 670 may be arbitrarily placed in one or more positions and/or orientations such that multiple magnetic fields are applied to the diamond having an unknown orientation 100′. Measurements of the fluorescence intensity response are taken for each position and/or orientation of the first magnetic field generator 670. Once a fluorescence intensity response 800 is produced that adequately separates out the four Lorentzian pairs, the position of the first magnetic field generator 670 is maintained and the process may proceed to calibration tests. In other embodiments, the separation process may be performed by the second magnetic field generator 675. In this case, the controller 680 may be configured to control the second magnetic field generator 675 to generate multiple magnetic fields until the desired separation is produced. In yet other embodiments, the first and/or second magnetic field generators may be affixed to a pivot assembly (e.g., a gimbal assembly) that may be controlled to hold and position the first and/or second magnetic field generators to a predetermined and well-controlled set of orientations, thereby establishing the desired Lorentzian separation and/or calibration magnetic fields (described below). In this case, the controller 680 may be configured to control the pivot assembly having the first and/or second magnetic field generators to position and hold the first and/or second magnetic field generators at the predetermined orientation.

After an appropriate calibration b_(bias) field has been found that adequately separates out the four Lorentzian dips, a measurement vector m_(bias) of the corresponding bias magnet's magnetic field projections is collected. The measurement vector may be expressed in a similar manner as the linear model described in equation (5): m _(bias) =|A ^(T) b _(bias) +n _(bias)|  (15)

As noted above with regard to equation (5), the variables represented in equation (15) are the same, but represented in relation to the applied bias field.

At this point, it is unknown which of the four Lorentzian dips correspond to which of the sensor axes, which still remain unknown. However, because any possible permutation of the axes' ordering can be captured by applying an appropriate orthogonal matrix to A_(S), and, because the process described herein is estimating the orthogonal matrix that best represents the data, any permutation of the axes' ordering will be compensated by the transformation. Due to this, the axes may be generally assigned such as, for example, the Lorentzian dip that is closest to the zero-field splitting frequency is assigned as a₁, the second-closest is assigned as a₂, and so on.

Sign Recovery of Magnetic Field Projections

Due to the symmetry of the sensor measurements, the obtained m_(bias) vector has no inherent sign information for each of its four components. However, sign information may be recovered using the following process.

The projections of the magnetic field vector onto the four axes is given by the vector A^(T)b. The sum of the projections may then be initially presumed to equal zero per the following: Σ_(i=1) ⁴(A ^(T) b)_(i)=Σ_(i=1) ⁴((RA _(S))^(T) b)_(i) =Σ_(i=1) ⁴ a _(S,i) ^(T) R ^(T) b =b ^(T) RΣ _(i=1) ⁴ a _(S,i) =b ^(T) RO =0  (16)

In the above equation (16), 0ε

^(4×1) represents a vector consisting of all zeros. If the sum of the elements of a vector xε

^(4×1) equals zero, then a magnetic field vector b may be found whose projections onto the four axes of a diamond is identical to x. In this regard, the magnetic field vector b may be defined as follow: b=0.75Ax  (17)

The projection of the magnetic field vector b onto the four axes of a diamond may be given by: A ^(T) b=0.75A ^(T) Ax =0.75(RA _(S))^(T) RA _(S) x =0.75A _(S) ^(T) R ^(T) RA _(S) x =0.75A _(S) ^(T) A _(S) x  (18)

The values for the A_(S) matrix from equations (1)-(2) may be plugged into equation (18) to give:

$\begin{matrix} {{A^{T}b} = {{\begin{bmatrix} {0.75} & {{- 0.}25} & {{- 0.}25} & {- {0.25}} \\ {{- 0.}25} & {0.75} & {{- 0.}25} & {{- 0.}25} \\ {{- 0.}25} & {{- 0.}25} & {0.75} & {{- 0.}25} \\ {{- 0.}25} & {{- 0.}25} & {{- 0.}25} & {0.75} \end{bmatrix}x} = {{\left( {I - {0.25\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{bmatrix}}} \right)x} = {x - {0.25\begin{bmatrix} {\sum\limits_{i = 1}^{4}x_{i}} \\ {\sum\limits_{i = 1}^{4}x_{i}} \\ {\sum\limits_{i = 1}^{4}x_{i}} \\ {\sum\limits_{i = 1}^{4}x_{i}} \end{bmatrix}}}}}} & (19) \end{matrix}$

Because it was initially assumed that the sum of all the elements of x equals 0, equation (19) can be reduced to:

$\begin{matrix} {{A^{T}b} = {{x - {0.25\begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}}} = x}} & (20) \end{matrix}$

Thus, a b vector exists whose projections onto the axes of a diamond is identical to x and the initial presumption of equation (16) is proved. Accordingly, the sum of the axes' projections of any magnetic field impinging on a diamond will be equal to zero, and measurements obtained, in the absence of noise, will sum to zero as well. Thus, sign information for the bias measurements may be recovered following this basic principle. This particular step is especially applicable if the bias magnetic field's projections are much larger than the expected noise levels.

With reference to FIG. 9, a method to recover sign information from the bias field measurements according to one embodiment will now be described. First, in a step S10, the largest of the four measurements is arbitrarily set to a sign value, either positive or negative. Once this is chosen, the next steps are dictated based on this sign choice such that the principles of equation (16) are maintained. For example, as shown in the embodiment of FIG. 8, the largest of the four measurements, measurement 810 a, is assigned as positive. Next, in a step S11, the second-largest measurement (e.g., measurement 810 b shown in FIG. 8) is set to negative. By setting the second-largest measurement to negative, the positive value assigned to the largest measurement may be offset toward zero. In a step S12, the third-largest measurement (e.g., measurement 810 c of FIG. 8) is assigned a negative sign value. Because, by definition, the second-largest measurement is smaller than the largest measurement, a negative sign value for the third-largest measurement will offset the largest measurement further towards zero. Finally, in a step S13, the smallest measurement is assigned either a positive or negative value that allows for the sum total of the four measurements to approximately equal zero. In FIG. 8, the smallest measurement 810 d is assigned a positive value. After this process, therefore, an appropriately signed m_(bias) vector may be obtained.

After application of the bias field that cleanly separates out the four Lorentzian dips and a measurement of the resulting bias field has been collected, a series of calibration tests may be performed. As shown in FIG. 10, a series of p known external magnetic fields, in conjunction with the fixed b_(bias) field, is applied and the resulting sensor measurements are collected. In some embodiments, a series of at least three p (p≧3) weak magnetic fields are applied. In particular embodiments, at least three non-coplanar p weak magnetic fields are applied. In yet other embodiments, three orthogonally spaced p (p≧3) weak magnetic fields are applied. In particular embodiments, four to five p (p=4, 5, . . . ) weak magnetic fields are applied. Such fields may be applied by the second magnetic generator 675 and, thus, controlled by the controller 680. The known applied external magnetic fields may be represented by the following matrix: B=[b ₁ b ₂ . . . b _(p)]  (21)

In equation (21), b_(k) represents the k^(th) field for k=1 . . . p. The obtained measurements m_(k) corresponding to each b_(k) may be represented by the linear model described above as: m _(k) =|A ^(T)(b _(k) +b _(bias))+n _(k)|  (22)

The portion of m_(k) that corresponds solely to the external magnetic field b_(k) can be isolated, along with proper sign values, by: {tilde over (m)} _(k)=(m _(k) −|A ^(T) b _(bias)|)∘sgn(A ^(T) b _(bias))  (23)

In the above equation, ∘ represents the Hadamard (i.e., element-wise) matrix product, while sgn( ) represents the element-wise signum function. At this stage, A^(T) remains unknown. However, A^(T)b_(bias) may be estimated. This is possible by substituting {hacek over (m)}_(bias) for A^(T)b_(bias) in equation (23): {tilde over (m)} _(k)≈(m _(k) −|{hacek over (m)} _(bias)|)∘sgn({hacek over (m)} _(bias))  (24)

Combining equations (22) and (23), the derived calibration measurement can be written as follows: {tilde over (m)} _(k) =A ^(T) b _(k) +ñ _(k)  (25)

In the above equation (25), ñ_(k)=n_(k)∘sgn({hacek over (m)}_(bias))+n_(bias).

By defining the matrices {tilde over (M)}=[{tilde over (m)}₁ {tilde over (m)}₂ . . . {tilde over (m)}_(p)] and Ñ=ñ₁ ñ₂ . . . ñ_(p)], the external magnetic fields and their corresponding measurements may be compactly represented by:

$\begin{matrix} {{{A^{T}\begin{bmatrix} b_{1} & b_{2} & ... & b_{p} \end{bmatrix}} + \begin{bmatrix} {\overset{\sim}{n}}_{1} & {\overset{\sim}{n}}_{2} & ... & {\overset{\sim}{n}}_{p} \end{bmatrix}} = {\left. \begin{bmatrix} {\overset{\sim}{m}}_{1} & {\overset{\sim}{m}}_{2} & ... & {\overset{\sim}{m}}_{p} \end{bmatrix}\mspace{20mu}\Rightarrow{{A^{T}B} + \overset{\sim}{N}} \right. = {\left. \overset{\sim}{M}\mspace{20mu}\Rightarrow{{\left( {RA}_{S} \right)^{T}B} + \overset{\sim}{N}} \right. = \overset{\sim}{M}}}} & (26) \end{matrix}$

Once the known B and the measured {tilde over (M)} have been obtained, equation (26) may be expanded as follows:

$\begin{matrix} {{{\left( {RA}_{S} \right)^{T}B} + \overset{\sim}{N}} = {\left. \overset{\sim}{M}\Rightarrow{{A_{S}^{T}R^{T}B} + \overset{\sim}{N}} \right. = {\left. \overset{\sim}{M}\Rightarrow{{A_{S}A_{S}^{T}R^{T}B} + {A_{S}\overset{\sim}{N}}} \right. = {\left. {A_{S}\overset{\sim}{M}}\Rightarrow{{\frac{4}{3}{IR}^{T}B} + {A_{S}\overset{\sim}{N}}} \right. = {\left. {A_{S}\overset{\sim}{M}}\Rightarrow{{R^{T}B} + {\frac{3}{4}A_{S}\overset{\sim}{N}}} \right. = {\frac{3}{4}A_{S}\overset{\sim}{M}}}}}}} & (27) \end{matrix}$

From equation (19), A_(S)A_(S) ^(T)=4/3I was demonstrated and thus substituted into equation (27) above. Because the singular values of A_(S) are known and equal (i.e., about 1.15), the noise term Ñ will not be colored or largely amplified in the expression 3/4A_(S)Ñ. Thus, we can treat the expression 3/4A_(S)Ñ as a new noise term:

$\begin{matrix} {\overset{\approx}{N} = {\frac{3}{4}A_{S}\overset{\sim}{N}}} & (28) \end{matrix}$

Combining equations (27) and (28) results in:

$\begin{matrix} {{{R^{T}B} + \overset{\approx}{N}} = {\frac{3}{4}A_{S}\overset{\sim}{M}}} & (29) \end{matrix}$

Taking the transpose of both sides of equation (29) gives:

$\begin{matrix} {{{B^{T}R} + {\overset{\approx}{N}}^{T}} = {\frac{3}{4}{\overset{\sim}{M}}^{T}A_{S}^{T}}} & (30) \end{matrix}$

In the next step, an orthogonal matrix {circumflex over (R)} is desired that provides the least-squares fit between B^(T) and 3/4{tilde over (M)}^(T)A_(S) ^(T) in equation (30). Some least-squares formulations may introduce translation and/or angular error into the orthogonal matrix {circumflex over (R)}. For example, error may be introduced when applying the matrix {circumflex over (R)} to the standard orientation matrix A_(S) in the form of a translation of the center of the axes from the standard orientation to the estimated orientation or in a change in the angles shown in equation (3) between given axes. Thus, a least-squares fit that can substantially maintain the relative orientation of the axes to each other when rotating from the standard orientation to the estimated orientation is preferable. In this regard, the orthogonal matrix may be expressed as: {circumflex over (R)}=argmin_(Rε0(3)) ∥B ^(T) R−3/4{tilde over (M)} ^(T) A _(S) ^(T)∥_(F)  (31)

Where, in equation (31), 0(3) represents the group of orthogonal 3×3 matrices and ∥ ∥_(F) represents the Frobenius norm.

By defining the orthogonal matrix {circumflex over (R)} as above, the particular problem may be reduced to the Orthogonal Procrustes Problem to solve for {circumflex over (R)}. First, the following is defined: Z=3/4B{tilde over (M)} ^(T) A _(S) ^(T)  (32)

A singular devalue decomposition of Z is performed to obtain: Z=UΣV ^(T)  (33)

Where in equation (33), U is an orthogonal 3×3 matrix that contains the left singular vectors of Z; Σ is an orthogonal 3×3 matrix that contains the singular values of Z; and V^(T) is an orthogonal 3×3 matrix that contains the right singular vectors of Z. Given the above, the solution to the Orthogonal Procrustes Problem of (33) is given by: {circumflex over (R)}=UV ^(T)  (34)

Accordingly, with equation (34), an estimate {circumflex over (R)} is obtained that may be applied to the standard orientation matrix A_(S) to give the true axes orientation matrix A. Thus, an estimate Â of A can be obtained by applying equation (4) to yield: Â={circumflex over (R)}A _(S)  (35)

In the embodiment described above, the Orthogonal Procrustes Problem provides an advantage in reducing translation and/or angular error that may be introduced by the least-squares fit and, thus, provides an accurate estimation of the needed rotation matrix. By accurately estimating the rotation matrix, an accurate estimation of the orientation of an arbitrarily placed lattice structure in a magnetic field detection system having a magneto-optical defect center material is produced. This, in turn, reduces the process to determining the orientation of a diamond in the magnetic detection system 600 to a simple calibration method that may be calculated and controlled by the controller 680 and performed before sensing begins, without the need for pre-manufacturing processes to orient the lattice structure relative to the sensor or additional equipment for visual aid inspection. Moreover, with the above, an accurate estimate of the true orientation of the axes of the NV diamond material 620 may be obtained and recovery of the external magnetic field for magnetic sensing, described further below, may be improved.

Once the axes have been determined using embodiments described above, the bias magnet's magnetic field can subsequently be optimally re-oriented using the methods described below along with the newfound knowledge of the axes' orientations.

Application of the Bias Magnetic Field

Once the orientation of the axes of the diamond lattice has been determined, a bias magnetic field may be applied to cleanly separate out the Lorenztian dips and obtain sign estimates of the magnetic field projections onto the identified diamond lattices.

As noted above, the baseline set of microwave resonance frequencies is defined as those frequencies which are created when no external magnetic field is present. When no bias magnet or bias coil is present (i.e., no bias magnet or bias coil is added internally to the system by, for example, the first and second magnetic field generators 670, 675), the baseline resonance frequencies will be identical for all four diamond axes (e.g., all approximately equal to 2.87 GHz). If a bias magnet or coil is introduced (e.g., applied by the first magnetic field generator 670 and/or second magnetic generator 675), the four axes' baseline resonance frequencies may be uniquely shifted if the projection of the bias magnet's magnetic field onto each of the four axes is unique. By applying a known bias magnetic field, the magnitude and orientation of a non-zero external magnetic field may then be determined by evaluating the additional shift in each axis' microwave resonance frequency relative to the baseline frequency offset, which will be described in more detail below.

For an external magnetic field in the absence of a bias magnetic field, the Lorentzian dips in the microwave resonance spectra that correspond to each of the four axes may overlap significantly. Such overlap can occur when either the projection of the external field onto multiple axes is similar, or when the width of the Lorentzian dips is much larger than the difference in the resonance frequency shifts due to the external magnetic field. In these cases, an external bias magnet applied as part of the system 600 may be used to minimize the overlap by significantly separating the Lorentzian spectral dips, thereby enabling unique recovery of the external magnetic field projections on each of the axes.

The following will describe how an optimal bias magnetic field via the first magnetic field generator 670 (e.g., a permanent magnet) and/or the second magnetic field generator 675 (e.g., three-axis Helmholtz coil system) is calculated by the controller 680 according to one embodiment. Once the optimal bias magnetic field is determined, the orientation of the bias magnet's magnetic field relative to the diamond may then be determined to produce the desired baseline shifts.

Similar to above when determining the orientation of the axes of the diamond lattice, the magnetic field generated by the bias magnet (e.g., the first magnetic field generator 670 and/or the second magnetic field generator 675) may be represented by the vector b_(bias)ε

^(3×1). As noted above, the projection of the bias magnetic field onto each of the four axes of the diamond is given by A^(T)b_(bias). The shifted baseline set of microwave resonance frequencies f relative to the centered zero-field splitting frequency (e.g., about 2.87 GHz) may be given by: f={±γ(A ^(T) b _(bias))₁, ±γ(A ^(T) b _(bias))₂, ±γ(A ^(T) b _(bias))₃, ±γ(A ^(T) b _(bias))₄,} ={±γa ₁ ^(T) b _(bias) , ±γa ₂ ^(T) b _(bias) , ±γa ₃ ^(T) b _(bias) , ±γa ₄ ^(T) b _(bias),}  (36)

In equation (36), it is noted that γ represents the nitrogen vacancy gyromagnetic ratio of about 28 GHz/T.

Depending on the characteristics of the sensor and its particular application, optimum performance of the sensor may be achieved under different sets of baseline frequencies. However, not all arbitrary baseline frequency sets may be realizable. Thus, the criteria for producing baseline offsets may be determined from which the corresponding required bias field may be computed.

First, f may be defined to represent the desired baseline set of microwave resonance frequencies relative to the centered zero-field splitting frequency and be expressed as follows: f={±f ₁ , ±f ₂ , ±f ₃ , ±f ₄,}  (37)

Using equation (36), if a b_(bias) exists that produces f, then the projections of b_(bias) onto the four axes of the diamond may be given by: a _(i) ^(T) b _(bias) =−fi/γ or fi/γ for i=1 . . . 4  (38)

Regardless of the sign value of the axis projection (i.e., whether positive or negative), the same pair of microwave resonance frequencies {−f_(i), f_(i)} will be produced by the system 600. Thus, there is freedom to choose whether each axis projection will be positive or negative without affecting the resulting baseline.

To confirm whether a b_(bias) actually exists that produces f, the concept that a bias field b_(bias) will exist only if the projections of b_(bias) onto the four diamond axes sum to zero is applied, which was shown above as true in equations (16)-(20). Thus, this concept may be expressed as: Σ_(i=1) ⁴ =a _(i) ^(T) b _(bias)=Σ_(i=1) ⁴ s _(i) fi/γ=0, where s _(1,2,3,4)ε{−1,1}  (39)

Accordingly, if a set {s₁, s₂, s₃, s₄} can be found that causes the sum in equation (39) to be zero, a b_(bias) vector will exist that produces the desired baseline f. From equation (17) above, b_(bias) may then be given by:

$\begin{matrix} {b_{bias} = {0.75{A\begin{bmatrix} {s_{1}\frac{f_{1}}{\gamma}} \\ {s_{2}\frac{f_{2}}{\gamma}} \\ {s_{3}\frac{f_{3}}{\gamma}} \\ {s_{4}\frac{f_{4}}{\gamma}} \end{bmatrix}}}} & (40) \end{matrix}$

Once that set {s₁, s₂, s₃, s₄} has been determined that results in a b_(bias) vector that produces the desired baseline f, the Lorentzian dips may be fine-tuned to a desired separation by applying the appropriate bias field using the first magnetic field generator 670 and/or the second magnetic field generator 675. For example, equal separation between each pair of adjacent dips in the microwave resonance spectra may be represented by the following baseline set: f={±α, ±3α, ±5α, ±7α,}  (41)

The above equation holds for any αε

. The separation between any pair of adjacent dips is 2α. In addition, a possible set {s₁, s₂, s₃, s₄} that results in the sum of projections summing to zero is {1, −1, −1, 1}. Thus, from equations (40) and (41), the b_(bias) that will produce an equally separated baseline set is given by:

$\begin{matrix} {b_{bias} = {0.75{A\begin{bmatrix} \frac{\alpha}{\gamma} \\ {- \frac{3\alpha}{\gamma}} \\ {- \frac{5\alpha}{\gamma}} \\ \frac{7\alpha}{\gamma} \end{bmatrix}}}} & (42) \end{matrix}$

Assuming that the diamond is in a standard orientation with respect to the coordinate reference frame (i.e., A=A_(S)), equation (42) will reduce to:

$\begin{matrix} {b_{bias} = {\frac{\alpha\sqrt{3}}{\gamma}\begin{bmatrix} 1 \\ 2 \\ 4 \end{bmatrix}}} & (43) \end{matrix}$

Alternatively, however, the b_(bias) may also be determined after the true axes orientation has been estimated using the methods described above. For example, the b_(bias) that will produce an equally separated baseline set for an arbitrarily orientated diamond will be given by substitution of equation (35) into equation (42) to yield:

$\begin{matrix} {b_{bias} = {0.75\hat{R}{A_{S}\begin{bmatrix} \frac{\alpha}{\gamma} \\ {- \frac{3\alpha}{\gamma}} \\ {- \frac{5\alpha}{\gamma}} \\ \frac{7\alpha}{\gamma} \end{bmatrix}}}} & (44) \end{matrix}$

Maximum separation or a single axis pair of dips in the microwave resonance spectra may also be achieved. The maximum separation may be represented by the following baseline set: f={±α, ±α, ±α, ±3α,}  (45)

The above equation holds for any αε

. The separation between the primary pair and the three other peak pairs of adjacent dips will be 2α. As described above, a possible set {s₁, s₂, s₃, s₄} that results in the sum of projections summing to zero is {−1, −1, −1, 1}. Thus, from equations (40) and (45), the b_(bias) that will produce a maximum single dip separated baseline set is given by:

$\begin{matrix} {b_{bias} = {0.75{A\begin{bmatrix} {- \frac{\alpha}{\gamma}} \\ {- \frac{\alpha}{\gamma}} \\ {- \frac{\alpha}{\gamma}} \\ \frac{3\alpha}{\gamma} \end{bmatrix}}}} & (46) \end{matrix}$

Assuming that the diamond is in a standard orientation with respect to the coordinate reference frame (i.e., A=A_(S)), equation (46) will reduce to:

$\begin{matrix} {b_{bias} = {\frac{4\alpha\sqrt{3}}{\gamma}\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}}} & (47) \end{matrix}$

It should be noted that equation (47) corresponds directly to one of the four axis orientations, α₄.

Alternatively, the b_(bias) may also be determined after the true axes orientation has been estimated using the methods described above. For example, the b_(bias) that will produce a maximum separated baseline set for an arbitrarily orientated diamond will be given by substitution of equation (35) into equation (46) to yield:

$\begin{matrix} {b_{bias} = {0.75\hat{R}{A_{S}\begin{bmatrix} {- \frac{\alpha}{\gamma}} \\ {- \frac{\alpha}{\gamma}} \\ {- \frac{\alpha}{\gamma}} \\ \frac{3\alpha}{\gamma} \end{bmatrix}}}} & (48) \end{matrix}$ Measuring an External Magnetic Field

Once a bias magnet and/or coil with a known bias magnetic field has been applied to the system 600 using the first and/or second magnetic field generators 670, 675 to produce a desired baseline set of microwave resonance frequencies, the magnitude and direction of an external magnetic field may be measured. By defining the external magnetic field at the location of the diamond sensor as b_(ext)ε

^(3×1), equation (5) may be expressed as: m=|A ^(T) b+n|=|A ^(T)(b _(ext) +b _(bias))+n|  (49)

The portion of m that corresponds to the external magnetic field may be isolated by comparing m to the known projections of the bias magnetic field b_(bias), which can be expressed as: m _(ext)=(m−|A ^(T) b _(bias)|)∘sgn(A ^(T) b _(bias))  (50)

In equation (50), ∘ denotes the Hadamard (element-wise) matrix product. The resulting m_(ext) will have the appropriate sign for the projection of b_(ext) onto each axis, thereby allowing unambiguous recovery of b_(ext) using the approach shown in equations (5)-(13), where m_(ext) is used in place of m to estimate b_(ext).

Based on the above, an unknown external magnetic field vector may be accurately estimated and recovered. FIG. 11 shows a flowchart illustrating a method for the recovery of an external magnetic field vector as implemented by the controller 680 of the system 600 using the methods described above. In a step S100, the bias magnetic field that will produce the desired separation between the Lorentzian responses for each diamond axis is computed using the methods described above (e.g., equal separation or maximum separation computations). Once this is determined, the first magnetic field generator 670 (e.g., a permanent magnet) may be positioned to produce the desired field or the second magnetic field generator 675 (e.g., three-axis Helmholtz coil) may be controlled by the controller 680 to generate the desired field. Next, in a step S200, a relative direction (i.e., sign value) is assigned to each Lorentzian pair using the sign recovery method described above and shown in FIG. 9.

Once the Lorentzian responses have been optimally separated by the application of an appropriate bias field and sign values of the pairs have been assigned, measurement data of the total magnetic field impinging on the system 600 is collected in a step S300. Then, in a step S400, shifts in the Lorentzian dips due to the external magnetic field are detected and computed based on the linear relationship between the application of the magnetic field vector projected on a given diamond axis and the resulting energy splitting between the m_(s)=−1 spin state and the m_(s)=+1 spin state. In a step S500, this shift information is then used along with the methods described using equations (49)-(50) to compute an estimate of the external magnetic field b_(ext).

The embodiments of the inventive concepts disclosed herein have been described in detail with particular reference to preferred embodiments thereof, but it will be understood by those skilled in the art that variations and modifications can be effected within the spirit and scope of the inventive concepts. 

What is claimed is:
 1. A system for magnetic detection of an external magnetic field, comprising: a nitrogen vacancy (NV) diamond material comprising a plurality of NV centers; a magnetic field generator configured to generate a magnetic field; a radio frequency (RF) excitation source configured to provide RF excitation to the NV diamond material; an optical excitation source configured to provide optical excitation to the NV diamond material; an optical detector configured to receive an optical signal emitted by the NV diamond material, the optical signal being a fluorescence intensity having a plurality of reduced responses across a frequency range of the RF excitation; and a controller configured to: calculate a control magnetic field that separates the plurality of reduced responses in the optical signal emitted by the NV diamond material; control the magnetic field generator to generate the control magnetic field; receive a light detection signal from the optical detector based on the optical signal emitted by the NV diamond material due to the sum of the generated control magnetic field and the external magnetic field; store measurement data based on the received light detection signal; and calculate a vector of the external magnetic field based on the stored measurement data.
 2. The system of claim 1, wherein the controller calculates a control magnetic field that equally separates the plurality of reduced responses.
 3. The system of claim 1, wherein the controller calculates a control magnetic field that maximally separates the plurality of reduced responses.
 4. The system of claim 1, wherein the controller is further configured to calculate an orientation of the NV diamond material.
 5. The system of claim 4, wherein the controller is configured to calculate the orientation of the NV diamond material based on a rotation and/or reflection of a standard orientation of the NV diamond material.
 6. The system of claim 1, wherein the controller is configured to assign a plurality of sign values to the measurement data such that a sum of the measurement data approaches zero.
 7. The system of claim 6, wherein the controller is configured to assign a positive sign value to the largest and smallest measurement data, and a negative sign value to the second-largest and third-largest measurement data.
 8. The system of claim 6, wherein the controller is configured to assign a positive sign value to the largest measurement data, and a negative sign value to the second-largest, third-largest, and smallest measurement data.
 9. The system of claim 1, further comprising a pivot assembly, wherein the magnetic field generator is affixed to the pivot assembly, the pivot assembly is configured to position the magnetic field generator to a predetermined orientation such that the magnetic field generator generates the calculated control magnetic field, and the controller is further configured to control the pivot assembly.
 10. A system for magnetic detection of an external magnetic field, comprising: a nitrogen vacancy (NV) diamond material comprising a plurality of NV centers; a radio frequency (RF) excitation source configured to provide RF excitation to the NV diamond material; an optical excitation source configured to provide optical excitation to the NV diamond material; an optical detector configured to receive an optical signal emitted by the NV diamond material, the optical signal being a fluorescence intensity having a plurality of reduced responses across a frequency range of the RF excitation; a magnetic field generator configured to generate a control magnetic field, the control magnetic field being configured to separate the plurality of reduced responses in the optical signal emitted by the NV diamond material; and a controller configured to: receive a light detection signal from the optical detector based on the optical signal emitted by the NV diamond material due to the sum of the control magnetic field and the external magnetic field; store measurement data based on the received light detection signal; and calculate a vector of the external magnetic field based on the stored measurement data.
 11. The system of claim 10, wherein the control magnetic field is configured to equally separate the plurality of reduced responses.
 12. The system of claim 10, wherein the control magnetic field is configured to maximally separate the plurality of reduced responses.
 13. The system of claim 10, wherein the controller is further configured to calculate an orientation of the NV diamond material.
 14. The system of claim 13, wherein the controller is configured to calculate the orientation of the NV diamond material based on a rotation and/or reflection of a standard orientation of the NV diamond material.
 15. The system of claim 10, wherein the magnetic field generator is a permanent magnet.
 16. The system of claim 15, further comprising a pivot assembly, wherein the magnetic field generator is affixed to the pivot assembly, the pivot assembly is configured to position the magnetic field generator to a predetermined orientation such that the magnetic field generator generates the control magnetic field, and the controller is further configured to control the pivot assembly.
 17. A system for magnetic detection of an external magnetic field, comprising: a nitrogen vacancy (NV) diamond material comprising a plurality of NV centers; a radio frequency (RF) excitation source configured to provide RF excitation to the NV diamond material; an optical excitation source configured to provide optical excitation to the NV diamond material; an optical detector configured to receive an optical signal emitted by the NV diamond material, the optical signal being a fluorescence intensity having a plurality of reduced responses across a frequency range of the RF excitation; a magnetic field generator affixed to a pivot assembly, the pivot assembly being configured to position the magnetic field generator to a predetermined orientation such that the magnetic field generator generates a control magnetic field that separates the plurality of reduced responses in the optical signal emitted by the NV diamond material; and a controller configured to: control the pivot assembly to position the magnetic field generator to the predetermined orientation to generate the control magnetic field; receive a light detection signal from the optical detector based on the optical signal emitted by the NV diamond material due to the sum of the control magnetic field and the external magnetic field; store measurement data based on the received light detection signal; and calculate a vector of the external magnetic field based on the stored measurement data.
 18. The system of claim 17, wherein the control magnetic field is configured to equally separate the plurality of reduced responses.
 19. The system of claim 17, wherein the control magnetic field is configured to maximally separate the plurality of reduced responses.
 20. The system of claim 17, wherein the controller is configured to calculate an orientation of the NV diamond material based on a rotation and/or reflection of a standard orientation of the NV diamond material. 